General Relativity via differential forms -- explorations in Plebanski's Formalism for GR

TL;DR

This thesis explores general relativity through chiral and Plebanski's formulations, revealing new structures and tools for analytical, numerical, and geometric analysis of Einstein's equations.

Contribution

It develops the geometric foundations, analyzes linearized and nonlinear equations, and investigates numerical evolution schemes within Plebanski's chiral framework.

Findings

Exposes additional structure in Einstein's equations.

Develops gauge fixing techniques for linearized equations.

Proposes evolution schemes based on chiral formulations.

Abstract

This thesis studies general relativity (GR) using chiral formulations, which take advantage of the decomposition of the four-dimensional Lorentz group into self-dual and anti-self-dual sectors. Within this framework, GR can be expressed using Plebanski's formulation, where the basic variables are triples of 2-forms rather than a metric, or alternatively through pure connection approaches. These viewpoints expose additional structure in Einstein's equations (EEs) and offer new analytical and numerical tools. Part I develops the geometric foundations using fibre bundles, where the 2-forms arise as soldering forms on an SO(3,C) bundle. Part II investigates the linearised form of EEs in the chiral setting, with particular attention to their gauge fixings. Part III extends this analysis to the nonlinear regime, and also examines the complex-geometric structure underlying black hole spacetimes. The final part turns to numerical relativity, exploring evolution schemes built from the chiral formulations and their associated gauge choices.